| L(s) = 1 | − 2-s + 3-s + 4-s + 1.83·5-s − 6-s − 7-s − 8-s + 9-s − 1.83·10-s − 0.198·11-s + 12-s + 14-s + 1.83·15-s + 16-s + 0.445·17-s − 18-s − 1.57·19-s + 1.83·20-s − 21-s + 0.198·22-s + 9.19·23-s − 24-s − 1.63·25-s + 27-s − 28-s − 9.88·29-s − 1.83·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.820·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.579·10-s − 0.0597·11-s + 0.288·12-s + 0.267·14-s + 0.473·15-s + 0.250·16-s + 0.107·17-s − 0.235·18-s − 0.360·19-s + 0.410·20-s − 0.218·21-s + 0.0422·22-s + 1.91·23-s − 0.204·24-s − 0.327·25-s + 0.192·27-s − 0.188·28-s − 1.83·29-s − 0.334·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 1.83T + 5T^{2} \) |
| 11 | \( 1 + 0.198T + 11T^{2} \) |
| 17 | \( 1 - 0.445T + 17T^{2} \) |
| 19 | \( 1 + 1.57T + 19T^{2} \) |
| 23 | \( 1 - 9.19T + 23T^{2} \) |
| 29 | \( 1 + 9.88T + 29T^{2} \) |
| 31 | \( 1 + 6.88T + 31T^{2} \) |
| 37 | \( 1 + 5.13T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 7.51T + 43T^{2} \) |
| 47 | \( 1 - 8.51T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 0.307T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 7.90T + 73T^{2} \) |
| 79 | \( 1 - 6.98T + 79T^{2} \) |
| 83 | \( 1 + 3.19T + 83T^{2} \) |
| 89 | \( 1 - 5.55T + 89T^{2} \) |
| 97 | \( 1 + 4.95T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.49068403065618155563513137887, −7.12078469304319977044284055681, −6.31841410640601667932074924635, −5.57587553284142770141569394303, −4.87243497855511456894876790140, −3.61828844920933141606611597551, −3.09603950672117069465681999398, −2.04193524717527508343077136088, −1.50532166815047045812582980333, 0,
1.50532166815047045812582980333, 2.04193524717527508343077136088, 3.09603950672117069465681999398, 3.61828844920933141606611597551, 4.87243497855511456894876790140, 5.57587553284142770141569394303, 6.31841410640601667932074924635, 7.12078469304319977044284055681, 7.49068403065618155563513137887
